Training while predicting on dataset
Yes, this method of training a model is commonly known as online learning and specific learning algorithms have been designed for this purpose, such as, Stochastic Gradient Descent(SGD). As opposed to Batch Gradient descent, which computes gradients over the entire training set at each step, the SGD algorithm computes gradients for individual samples and updates the model's parameters.
The online learning strategy is not specific to Reinforcement Learning(RL) methods but is also suitable for the broader classification of Machine Learning algorithms. Suppose, we are given a simple model pretrained on a large Face Recognition dataset, and wish the model to continuously adapt to new faces, we may keep feeding in the new input images with their correct labels (obtained from feedback), as they appear, and apply SGD to incrementally update the model. To go about this using an RL based approach, we may penalize the model with negative rewards for every label misclassified and update it.